Autocommutators and Engel groups

For a given group G, with an element x in G and automorphism a in Aut(G), the n-th autocommutator [x;n a] is defined recursively by [x;a] = x􀀀1xa and [x;n a] = [[x;n􀀀1 a];a] for all n 1. Th e group G is said to be n-auto-Engel if [x;n a] = [a;n x] = 1, for all x 2 G and all a 2 Aut(G), where [a;x] = [x;a]􀀀1. In the presented paper, we study the structure of 2-auto- Engel groups and show that these groups indeed satisfy the equation a(x)a􀀀1(x) = x2, for all x 2 G and a 2 Aut(G).